time
to
expiry
(
years
)
T
3
starting
asset
price
S
0
40
strike
price
X
45
risk

free
rate
r
0
.
07
dividend
yield
δ
0
volatility
σ
0
.
3
Geske

Johnson
approximation
for
the
American
put
=
8
.
06486
Bermudan
put
P
3
=
7
.
46865
Bermudan
put
P
2
=
7
.
16061
European
put
P
1
=
6
.
19676
32
33
34
35
36
37
38
S
*
T
/
2
6
.
95
7
.
00
7
.
05
7
.
10
7
.
15
P
2